Mechanics of Microsystems

Alberto Corigliano, Raffaele Ardito, Claudia Comi, Attilio Frangi, Aldo Ghisi, and Stefano Mariani – Politecnico di Milano, Italy Alberto Corigliano is a Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. A. Corigliano has authored and co-authored more than 240 scientific publications in fields related to solid and structural mechanics at various scales, including 2 book chapters in Microsystems area, and 7 patents on Microsystems. During his research activity, A. Corigliano covered a wide range of subjects in the fields of structural and materials mechanics, with particular reference to theoretical and computational problems relevant to non-linear material responses. Raffaele Ardito is an Associate Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. He graduated in 2000 (cum laude) at the Politecnico di Milano in Civil Engineering and he received the Ph.D. degree, cum laude, in 2004. From 2004 to 2006 he was a research fellow at the National Institute for Nuclear Physics, joining an international research group with focus on solid mechanics in cryogenic conditions. He spent, in 2008 and 2010, two periods of research at the Research Laboratory of Electronics, Massachusetts Institute of Technology, as visiting scientist. His scientific contributions to the field of MEMS focus on theoretical and computational aspects of adhesion and multi-physics behavior. Claudia Comi is a Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. C. Comi has authored and co-authored more than 140 scientific publications in various fields of solid and structural mechanics and 4 patents on Microsystems. Her main research interests concern theoretical and computational mechanics of materials and structures. Her research activities focus on damage and quasi-brittle fracture modelling, on instability phenomena and nonlocal models for elastoplastic and damaging one-phase and multi-phase materials, including functionally graded materials, and on design and reliability of MEMS. Attilio Frangi is a Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. A. Frangi has authored and co-authored more than 150 scientific publications on issues of computational mechanics and micromechanics and 5 patents on Microsystems. He has co-edited one scientific monograph on the multi-physics simulation of MEMS and NEMS. The research interests of A. Frangi in the field of MEMS include: the design of new devices; the theoretical and numerical analysis of multi-physics phenomena; the analysis of non-linear phenomena in the dynamical response of MOEMS. Aldo Ghisi is an Assistant Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. A. Ghisi has authored and co-authored more than 70 scientific publications on various subjects related to materials and structural mechanics. His research areas include multi-physics phenomena in micro/nano structures, particularly related to mechanical simulation of drop impacts, fatigue in polysilicon, gas-solid interaction, study of wafer-to-wafer bonding. Besides microsystems, he is also involved in the numerical and experimental study of metallic alloys for cryogenic applications and in dam engineering. Stefano Mariani is an Associate Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. S. Mariani has authored and co-authored about 170 scientific publications. His main research interests are: numerical simulations of ductile fracture in metals and quasi-brittle fracture in heterogeneous and functionally graded materials; extended finite element methods; calibration of constitutive models via extended and sigma-point Kalman filters; multi-scale solution methods for dynamic delamination in layered composites; reliability of MEMS subject to shocks and drops; structural health monitoring of composite structures through MEMS sensors.

• Series Preface xiiiPreface xvAcknowledgements xviiNotation xixAbout the Companion Websitexxiii1 Introduction 11.1 Microsystems 11.2 Microsystems Fabrication 31.3 Mechanics in Microsystems 51.4 Book Contents 6References 7Part I Fundamentals 92 Fundamentals of Mechanics and Coupled Problems 112.1 Introduction 112.2 Kinematics and Dynamics of Material Points and Rigid Bodies 122.2.1 Basic Notions of Kinematics and Motion Composition 122.2.2 Basic Notions of Dynamics and Relative Dynamics 152.2.3 One-Degree-of-Freedom Oscillator 172.2.4 Rigid-Body Kinematics and Dynamics 222.3 Solid Mechanics 252.3.1 Linear Elastic Problem for Deformable Solids 262.3.2 Linear Elastic Problem for Beams 352.4 Fluid Mechanics 432.4.1 Navier–Stokes Equations 432.4.2 Fluid–Structure Interaction 482.5 Electrostatics and Electromechanics 492.5.1 Basic Notions of Electrostatics 492.5.2 Simple Electromechanical Problem 542.5.3 General Electromechanical Coupled Problem 582.6 Piezoelectric Materials in Microsystems 602.6.1 Piezoelectric Materials 602.6.2 Piezoelectric Modelling 622.7 Heat Conduction and Thermomechanics 642.7.1 Heat Problem 642.7.2 Thermomechanical Coupled Problem 67References 703 Modelling of Linear and Nonlinear Mechanical Response 733.1 Introduction 733.2 Fundamental Principles 743.2.1 Principle of Virtual Power 743.2.2 Total Potential Energy Principle 743.2.3 Hamilton’s Principle 753.2.4 Specialization of the Principle of Virtual Powers to Beams 763.3 Approximation Techniques and Weighted Residuals Approach 763.4 Exact and Approximate Solutions for Dynamic Problems 793.4.1 Free Flexural Linear Vibrations of a Single-span Beam 793.4.2 Nonlinear Vibration of an Axially Loaded Beam 803.5 Example of Application: Bistable Elements 84References 90Part II Devices 914 Accelerometers 934.1 Introduction 934.2 Capacitive Accelerometers 944.2.1 In-Plane Sensing 944.2.2 Out-of-Plane Sensing 964.3 Resonant Accelerometers 984.3.1 Resonating Proof Mass 984.3.2 Resonating Elements Coupled to the Proof Mass 994.4 Examples 1014.4.1 Three-Axis Capacitive Accelerometer 1014.4.2 Out-of-Plane Resonant Accelerometer 1044.4.3 In-Plane Resonant Accelerometer 1054.5 Design Problems and Reliability Issues 107References 1075 Coriolis-Based Gyroscopes 1095.1 Introduction 1095.2 Basic Working Principle 1095.2.1 Sensitivity of Coriolis Vibratory Gyroscopes 1125.3 Lumped-Mass Gyroscopes 1135.3.1 Symmetric and Decoupled Gyroscope 1135.3.2 Tuning-Fork Gyroscope 1145.3.3 Three-Axis Gyroscope 1155.3.4 Gyroscopes with Resonant Sensing 1155.4 Disc and Ring Gyroscopes 1185.5 Design Problems and Reliability Issues 118References 1196 Resonators 1216.1 Introduction 1216.2 Electrostatically Actuated Resonators 1236.3 Piezoelectric Resonators 1256.4 Nonlinearity Issues 126References 1287 Micromirrors and Parametric Resonance 1317.1 Introduction 1317.2 Electrostatic Resonant Micromirror 1327.2.1 Numerical Simulations with a Continuation Approach 1367.2.2 Experimental Set-Up 140References 1458 Vibrating Lorentz Force Magnetometers 1478.1 Introduction 1478.2 Vibrating Lorentz Force Magnetometers 1488.2.1 Classical Devices 1488.2.2 Improved Design 1518.2.3 Further Improvements 1558.3 Topology or Geometry Optimization 156References 1599 Mechanical Energy Harvesters 1619.1 Introduction 1619.2 Inertial Energy Harvesters 1629.2.1 Classification of Resonant Energy Harvesters 1629.2.2 Mechanical Model of a Simple Piezoelectric Harvester 1659.3 Frequency Upconversion and Bistability 1749.4 Fluid–Structure Interaction Energy Harvesters 1769.4.1 Synopsis of Aeroelastic Phenomena 1779.4.2 Energy Harvesting through Vortex-Induced Vibration 1799.4.3 Energy Harvesting through Flutter Instability 180References 18110 Micropumps 18510.1 Introduction 18510.2 Modelling Issues for Diaphragm Micropumps 18610.3 Modelling of Electrostatic Actuator 18810.3.1 Simplified Electromechanical Model 18810.3.2 Reliability Issues 19210.4 Multiphysics Model of an Electrostatic Micropump 19610.5 Piezoelectric Micropumps 19810.5.1 Modelling of the Actuator 19810.5.2 Complete Multiphysics Model 201References 202Part III Reliability and Dissipative Phenomena 20511 Mechanical Characterization at the Microscale 20711.1 Introduction 20711.2 Mechanical Characterization of Polysilicon as a Structural Material for Microsystems 20911.2.1 Polysilicon as a Structural Material for Microsystems 20911.2.2 Testing Methodologies 21011.2.3 Quasi-Static Testing 21111.2.4 High-Frequency Testing 21411.3 Weibull Approach 21511.4 On-Chip Testing Methodology for Experimental Determination of Elastic Stiffness and Nominal Strength 21911.4.1 On-Chip Bending Test through a Comb-Finger Rotational Electrostatic Actuator 22011.4.2 On-Chip Bending Test through a Parallel-Plate Electrostatic Actuator 22511.4.3 On-Chip Tensile Test through an Electrothermomechanical Actuator 22911.4.4 On-Chip Test for Thick Polysilicon Films 233References 24012 Fracture and Fatigue in Microsystems 24512.1 Introduction 24512.2 Fracture Mechanics: An Overview 24512.3 MEMS Failure Modes due to Cracking 24912.3.1 Cracking and Delamination at Package Level 24912.3.2 Cracking at Silicon Film Level 25012.4 Fatigue in Microsystems 25612.4.1 An Introduction to Fatigue in Mechanics 25612.4.2 Polysilicon Fatigue 25912.4.3 Fatigue in Metals at the Microscale 26112.4.4 Fatigue Testing at the Microscale 263References 26613 Accidental Drop Impact 27113.1 Introduction 27113.2 Single-Degree-of-Freedom Response to Drops 27213.3 Estimation of the Acceleration Peak Induced by an Accidental Drop 27613.4 A Multiscale Approach to Drop Impact Events 27713.4.1 Macroscale Level 27713.4.2 Mesoscale Level 27913.4.3 Microscale Level 27913.5 Results: Drop-Induced Failure of Inertial MEMS 280References 28714 Fabrication-Induced Residual Stresses and Relevant Failures 29114.1 Main Sources of Residual Stresses in Microsystems 29114.2 The Stoney Formula and its Modifications 29214.3 Experimental Methods for the Evaluation of Residual Stresses 29914.4 Delamination, Buckling and Cracks in Thin Films due to Residual Stresses 304References 31015 Damping in Microsystems 31315.1 Introduction 31315.2 Gas Damping in the Continuum Regime with Slip Boundary Conditions 31415.2.1 Experimental Validation at Ambient Pressure 31715.2.2 Effects of Decreasing Working Pressure 31815.3 Gas Damping in the Rarefied Regime 32015.3.1 Evaluation of Damping at Low Pressure using Kinetic Models 32115.3.2 Linearization of the BGK Model 32315.3.3 Numerical Implementation 32415.3.4 Application to MEMS 32515.4 Gas Damping in the Free-Molecule Regime 32815.4.1 Boundary Integral Equation Approach 32815.4.2 Experimental Validations 33015.5 Solid Damping: Thermoelasticity 33515.6 Solid Damping: Anchor Losses 33815.6.1 Analytical Estimation of Dissipation 33915.6.2 Numerical Estimation of Anchor Losses 34215.7 Solid Damping: Additional unknown Sources – Surface Losses 34615.7.1 Solid Damping: Deviations from Thermoelasticity 34615.7.2 Solid Damping: Losses in Piezoresonators 346References 34816 Surface Interactions 35116.1 Introduction 35116.2 Spontaneous Adhesion or Stiction 35216.3 Adhesion Sources 35316.3.1 Capillary Attraction 35316.3.2 Van der Waals Interactions 35616.3.3 Casimir Forces 35816.3.4 Hydrogen Bonds 35916.3.5 Electrostatic Forces 36016.4 Experimental Characterization 36116.4.1 Experiments by Mastrangelo and Hsu 36116.4.2 Experiments by the Sandia Group 36216.4.3 Experiments by the Virginia Group 36516.4.4 Peel Experiments 36716.4.5 Pull-in Experiments 36816.4.6 Tests for Sidewall Adhesion 37216.5 Modelling and Simulation 37416.5.1 Lennard-Jones Potential 37416.5.2 Tribological Models: Hertz, JKR, DMT 37516.5.3 Computation of Adhesion Energy 37716.6 Recent Advances 38016.6.1 Finite Element Analysis of Adhesion between Rough Surfaces 38016.6.2 Accelerated Numerical Techniques 383References 387Index 393